It is shown in [1]} that such a generating function always exist. In free particle models, one obtains a form similar to (REF ), where the free energy \(F\left(\epsilon (p)\right)\)  controls the contribution of each mode, but with a measure determined by the energy \(E(p)\)  instead of the momentum. It is then natural to expect that this generalises to integrable models as
\(\mathcal {G} = \int \frac{{\rm d}E(p)}{2\pi } F\left(\epsilon (p)\right)\;.\) 
